Geodesic
HOD-supercompactness, Indestructibility, and Level by Level Equivalence
Arthur W. Apter 1   ; Shoshana Friedman 2
1 Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A. and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, NY 10016, U.S.A.
2 Department of Mathematics and Computer Science Kingsborough Community College-CUNY 2001 Oriental Blvd, Brooklyn, NY 11235, U.S.A.
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014) no. 3, pp. 197-209 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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In an attempt to extend the property of being supercompact but not HOD-supercompact to a proper class of indestructibly supercompact cardinals, a theorem is discovered about a proper class of indestructibly supercompact cardinals which reveals a surprising incompatibility. However, it is still possible to force to get a model in which the property of being supercompact but not HOD-supercompact holds for the least supercompact cardinal $\kappa _0$, $\kappa _0$ is indestructibly supercompact, the strongly compact and supercompact cardinals coincide except at measurable limit points, and level by level equivalence between strong compactness and supercompactness holds above $\kappa _0$ but fails below $\kappa _0$. Additionally, we get the property of being supercompact but not HOD-supercompact at the least supercompact cardinal, in a model where level by level equivalence between strong compactness and supercompactness holds.
DOI : 10.4064/ba62-3-1
Keywords: attempt extend property being supercompact hod supercompact proper class indestructibly supercompact cardinals theorem discovered about proper class indestructibly supercompact cardinals which reveals surprising incompatibility however still possible force get model which property being supercompact hod supercompact holds least supercompact cardinal kappa kappa indestructibly supercompact strongly compact supercompact cardinals coincide except measurable limit points level level equivalence between strong compactness supercompactness holds above kappa fails below kappa additionally get property being supercompact hod supercompact least supercompact cardinal model where level level equivalence between strong compactness supercompactness holds

Arthur W. Apter  1   ; Shoshana Friedman  2

1 Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A. and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, NY 10016, U.S.A.
2 Department of Mathematics and Computer Science Kingsborough Community College-CUNY 2001 Oriental Blvd, Brooklyn, NY 11235, U.S.A. 
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     author = {Arthur W. Apter and Shoshana Friedman},
     title = {HOD-supercompactness, {Indestructibility,
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     journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
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Arthur W. Apter; Shoshana Friedman. HOD-supercompactness, Indestructibility,
 and Level by Level Equivalence. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 62 (2014) no. 3, pp. 197-209. doi: 10.4064/ba62-3-1

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